Visualizing Tensor Normal Distributions at Multiple Levels of Detail

Abstract

Despite the widely recognized importance of symmetric second order tensor fields in medicine and engineering, the visualization of data uncertainty in tensor fields is still in its infancy. A recently proposed tensorial normal distribution, involving a fourth order covariance tensor, provides a mathematical description of how different aspects of the tensor field, such as trace, anisotropy, or orientation, vary and covary at each point. However, this wealth of information is far too rich for a human analyst to take in at a single glance, and no suitable visualization tools are available. We propose a novel approach that facilitates visual analysis of tensor covariance at multiple levels of detail. We start with a visual abstraction that uses slice views and direct volume rendering to indicate large-scale changes in the covariance structure, and locations with high overall variance. We then provide tools for interactive exploration, making it possible to drill down into different types of variability, such as in shape or orientation. Finally, we allow the analyst to focus on specific locations of the field, and provide tensor glyph animations and overlays that intuitively depict confidence intervals at those points. Our system is demonstrated by investigating the effects of measurement noise on diffusion tensor MRI, and by analyzing two ensembles of stress tensor fields from solid mechanics.

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Zusätzliches Material

Bibtex

@ARTICLE{abbasloo:VIS2015,
author = {Abbasloo, Amin and Wiens, Vitalis and Hermann, Max and Schultz, Thomas},
pages = {975--984},
title = {Visualizing Tensor Normal Distributions at Multiple Levels of Detail},
journal = {IEEE Trans. on Visualization and Computer Graphics},
volume = {22},
number = {1},
year = {2016},
keywords = {direct volume rendering, glyph based visualization., interaction, tensor visualization, Uncertainty
visualization},
abstract = {Despite the widely recognized importance of symmetric second order tensor fields in medicine and
engineering, the visualization of data uncertainty in tensor fields is still in its infancy. A
recently proposed tensorial normal distribution, involving a fourth order covariance tensor,
provides a mathematical description of how different aspects of the tensor field, such as trace,
anisotropy, or orientation, vary and covary at each point. However, this wealth of information is
far too rich for a human analyst to take in at a single glance, and no suitable visualization tools
are available. We propose a novel approach that facilitates visual analysis of tensor covariance at
multiple levels of detail. We start with a visual abstraction that uses slice views and direct
volume rendering to indicate large-scale changes in the covariance structure, and locations with
high overall variance. We then provide tools for interactive exploration, making it possible to
drill down into different types of variability, such as in shape or orientation. Finally, we allow
the analyst to focus on specific locations of the field, and provide tensor glyph animations and
overlays that intuitively depict confidence intervals at those points. Our system is demonstrated by
investigating the effects of measurement noise on diffusion tensor MRI, and by analyzing two
ensembles of stress tensor fields from solid mechanics.}
}